Vinegar is a preservative so it doesn't need to be preserved by refrigeration.

This begs a question - do any vinegars need to be refrigerated? I usually have white and red wine, sherry, rice wine, cider, balsamic and sometimes champagne or malt vinegars on hand - and it's never occurred to me to store them in the fridge.

I know the labels don't say to do it, but some things keep better refrigerated (e.g., nuts) even though not usually labelled that way. Vinegar?

First place to check for information is the label on the bottle. Unless it says "refrigerate after opening" you wouldn't have to. Vinegar is already/supposed to be spoiled, sour, fermented, so it won't go bad under normal conditions.

I don't refrigerate mine and it's fine.

No. Absolutely not.